4 ON CEETAIN SERIES OF SECTIONS OF THE 



cubes exactly till thrce-dimensionnl space and cannot thei'cfore form 

 a four-dimensional angle. Hence there cannot he a figure whose 

 angles are formed by eight cubes. Still less can there he one 

 whose angles are made by 20. Similar reasoning applies to the 

 dodecahedron ; as with the cubes, there is only room round a 

 point for four. Now the figure corresponding to the first case is 

 the S-cell, that corresponding to the second is the 120-cell. 



Taking an octahedron. A section by a s])ace cutting the edges 

 meeting at a vertex e(iually, is a square, and, as a cube is the 

 only regular three-dimensional figure bounded by squares, there 

 will only be one regular figure bounded by octahedra and that 

 will have six at each vertex. This is the 24-cell. 



In this manner we meet successivelv all the regvdar cells of 

 four-dimensional space. 



2. In making a seiies of sections of a regular four-dimensional 

 figure by three-dimensional spaces S , S , etc. parallel to a bounding 

 solid Z, that solid itself may be considered the first element 



of the series. If ^ be in a space S , it is the oidy part of the 

 figure in that space, but its faces are the surfaces of contact of it 

 with other bounding solids. So that in building up the solids of the 

 figure about Z in their position in four-dimensional space, if such 

 a thing were possible, there would be one on each face. In some 

 cases there are also one or more on each edge and one or more 

 on each vertex. Whether this be so or not, can be determined 

 by means of the number oï cells meeting at each vertex in the 

 particular figure under consideration. The solids on the faces of 

 Z may be sup])osed to turn about those faces until they lie wholly 

 in S and if there be any on the edges and vertices they may be 

 supposed to turn about those edges and vertices until they too lie 

 in jS . We represent in fig. 1 the result of such an operation 

 on the 8-cell. The cube JI A is the solid originally in the space 

 jS ; NA has been turned about its surface of contact with HA, 

 namely the square C A, into S . The cubes PA and SA have 

 been turned about the s([uares G ^l and E A respectively into the 

 same space S . 



The result of this is that the s(|uare L A, which is common to 

 the two cubes N A and P A, has assumed two positions in S . It 

 is e. g. horizontal in P A and vertical in N A. 



Similarly M A and A each appear in two positions in 8 . 



